Gauss–Jordan elimination

In linear algebra, Gauss–Jordan elimination is a version of Gaussian elimination that puts zeros both above and below each pivot element as it goes from the top row of the given matrix to the bottom. Every matrix has a reduced row echelon form, and both algorithms are guaranteed to produce it.

It is named after Carl Friedrich Gauss and Wilhelm Jordan, because it is a modification of Gaussian elimination as described by Jordan in 1887. However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss–Jordan elimination independently.

In computer science, Gauss–Jordan elimination as an algorithm has a time complexity of O(n³).

Application to finding inverses

If Gauss–Jordan elimination is applied on a square matrix, it can be used to calculate the matrix's inverse. This can be done by augmenting the square matrix with the identity matrix of the same dimensions, and through the following matrix operations:

If the original square matrix, A, is given by the following expression:

Then, after augmenting by the identity, the following is obtained:

By performing elementary row operations on the [AI] matrix until it reaches reduced row echelon form, the following is the final result:

The matrix augmentation can now be undone, which gives the following:

A matrix is non-singular (meaning that it has an inverse matrix) if and only if the identity matrix can be obtained using only elementary row operations.